$\tau$-Tilting Theory and $\tau$-Slices

TL;DR

This paper generalizes Brenner-Butler's tilting theorem within $ au$-tilting theory, introduces $ au$-slices, and explores their applications to tilted and cluster tilted algebras, including extensions.

Contribution

It extends classical tilting theory to support $ au$-tilting modules, introduces $ au$-slices, and applies these concepts to study various algebra extensions.

Findings

Complete slices of tilted algebras are $ au$-slices.

Local slices of cluster tilted algebras are $ au$-slices.

Extensions of algebras with $ au$-slices are studied.

Abstract

Comparing the module categories of an algebra and of the endomorphism algebra of a given support $\tau$-tilting module, we give a generalization of the Brenner-Butler's tilting theorem in the framework of $\tau$-tilting theory. Afterwards we define $\tau$-slices and prove that complete slices of tilted algebras and local slices of cluster tilted algebras are examples of complete $\tau$-slices. Then we apply this concept to the study of simply connected tilted algebras. Finally, we study the one-point extensions and the split-by-nilpotent extensions of an algebra with $\tau$-slices.